BitZone910/Trace-OPSD-Qwen2.5-3B
Text Generation • 3B • Updated • 219
Datasets:
Tasks:
Text Generation
Modalities:
Text
Formats:
parquet
Languages:
English
Size:
10K - 100K
License:
data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | problem stringlengths 13 2.2k | hindsight stringlengths 348 1.2k |
|---|---|---|---|---|---|---|
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find the least integer value of $x$ for which $2|x| + 7 < 17$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This process MUST ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"-4"
]
}
} | {
"index": 0,
"source_id": "numinamath-14",
"split": "train"
} | Find the least integer value of $x$ for which $2|x| + 7 < 17$. | - Recognize that the inequality contains an absolute value, so first rearrange it to isolate |x| on one side.
- Once the inequality has the form |x| < c, translate it into the equivalent open interval condition on x.
- Because the inequality is strict, endpoints of the interval are excluded; identify the boundary integ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let $(F_n)_{n\\in{N^*}}$ be the Fibonacci sequence defined by\r $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}$ for every $n\\geq{2}$ . Find\r\nthe limit: \\[ \\lim_{n \\to \\infty}(\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2"
]
}
} | {
"index": 1,
"source_id": "numinamath-16",
"split": "train"
} | Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \] | - Recognize the target as the limit of partial sums of a series whose terms pair Fibonacci numbers with a geometric factor.
- Generate the Fibonacci terms needed to evaluate the partial sum for increasing values of n, tracking the running total rather than simplifying immediately.
- Observe how the increment from each ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A positive integer is called a \"lucky number\" if it satisfies the following conditions:\n(1) It consists of only the digits \"6\" and \"8\";\n(2) It has at least two consecutive 8's... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"201"
]
}
} | {
"index": 2,
"source_id": "numinamath-17",
"split": "train"
} | A positive integer is called a "lucky number" if it satisfies the following conditions:
(1) It consists of only the digits "6" and "8";
(2) It has at least two consecutive 8's.
For example, 886 is a three-digit "lucky number", 8866 is a four-digit "lucky number", and 686888 is a six-digit "lucky number". How many eight... | - Interpret an eight-digit lucky number as a length-8 string formed only from the digits 6 and 8, so the entire search space is all sequences in {6,8}^8.
- Enumerate every such sequence systematically; since the length is small and the alphabet has only two symbols, a direct exhaustive generation will not miss any arra... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: There are $4$ different gifts. If they are all to be distributed to two people, A and B, with each person receiving at least one gift, the total number of different ways to distribute... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"14"
]
}
} | {
"index": 3,
"source_id": "numinamath-19",
"split": "train"
} | There are $4$ different gifts. If they are all to be distributed to two people, A and B, with each person receiving at least one gift, the total number of different ways to distribute the gifts is ______. | - Treat the gifts as distinct and the recipients A and B as labeled, so a distribution is determined by choosing the subset of gifts assigned to A; the remaining gifts go to B.
- Let k be the number of gifts assigned to A. The condition that both people receive at least one gift restricts k to the values from 1 through... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: The real-coefficient polynomials \\( f(x) \\) and \\( g(x) \\) satisfy \\( (1+\\mathrm{i} x)^{2001}=f(x)+\\mathrm{i} g(x) \\). Then the sum of the coefficients of \\( f(x) \\) is ____... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2^{1000}"
]
}
} | {
"index": 4,
"source_id": "numinamath-24",
"split": "train"
} | The real-coefficient polynomials \( f(x) \) and \( g(x) \) satisfy \( (1+\mathrm{i} x)^{2001}=f(x)+\mathrm{i} g(x) \). Then the sum of the coefficients of \( f(x) \) is _______. | - Recognize that the sum of coefficients of f(x) is obtained by evaluating f at x=1, so avoid expanding the large polynomial directly.
- Apply the same evaluation to both sides of the identity; since f and g have real coefficients, the desired quantity is the real part of the resulting complex number.
- Express the com... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find the number of subsets \\( S \\) of \\(\\{1,2, \\ldots 6\\}\\) satisfying the following conditions:\n- \\( S \\) is non-empty.\n- No subset of \\( S \\) has the property that the ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"34"
]
}
} | {
"index": 5,
"source_id": "numinamath-25",
"split": "train"
} | Find the number of subsets \( S \) of \(\{1,2, \ldots 6\}\) satisfying the following conditions:
- \( S \) is non-empty.
- No subset of \( S \) has the property that the sum of its elements is 10. | - Since the ground set {1,...,6} is small, start by enumerating all non-empty subsets S as candidates, using combinations or a bitmask representation.
- For each candidate S, translate the condition into a universal check: every non-empty subset of S must be tested for whether its elements sum to 10.
- Examine all subs... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given two sets $A=\\{1, 2, 3\\}$ and $B=\\{2, 3\\}$. Define the operation \"$*$\" between sets $A$ and $B$ as $A*B=\\{x | x=x_1+x_2, x_1 \\in A, x_2 \\in B\\}$. Determine the largest ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"15"
]
}
} | {
"index": 6,
"source_id": "numinamath-32",
"split": "train"
} | Given two sets $A=\{1, 2, 3\}$ and $B=\{2, 3\}$. Define the operation "$*$" between sets $A$ and $B$ as $A*B=\{x | x=x_1+x_2, x_1 \in A, x_2 \in B\}$. Determine the largest element in the set $A*B$ and the number of all proper subsets of $A*B$. | - Interpret the set operation as forming every possible sum of one element from A and one element from B, so the result consists of pairwise sums x1+x2.
- Because the result is a set, account for duplicate sums and count only distinct resulting values before proceeding.
- Find the largest value by considering the sums ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: If \\(2^{x}=15\\) and \\(15^{y}=32\\), the value of \\(xy\\) is:\n(A) 5\n(B) 8\n(C) 16\n(D) 6\n(E) 4\n\nNow it's your turn to respond for the current step.\nYou should first conduct a... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"5"
]
}
} | {
"index": 7,
"source_id": "numinamath-35",
"split": "train"
} | If \(2^{x}=15\) and \(15^{y}=32\), the value of \(xy\) is:
(A) 5
(B) 8
(C) 16
(D) 6
(E) 4 | - Notice the two equations share the same number 15, with 2 and 32 appearing as the other bases/results; this suggests combining the exponentials rather than solving each independently.
- Apply logarithms to both equations and use the power rule for logarithms to move the unknown exponents in front of the logarithms.
-... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find the largest 12-digit number \\( N \\) that satisfies the following two conditions:\n\n- The decimal representation of \\( N \\) contains six digits \"4\" and six digits \"7\";\n-... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"777744744744"
]
}
} | {
"index": 8,
"source_id": "numinamath-37",
"split": "train"
} | Find the largest 12-digit number \( N \) that satisfies the following two conditions:
- The decimal representation of \( N \) contains six digits "4" and six digits "7";
- In the decimal representation of \( N \), no four consecutive digits form the number "7444". | - Model the number as a length-12 string over the digits 4 and 7 with exactly six occurrences of each, avoiding the forbidden block "7444".
- Avoid enumerating all permutations; build the string left to right with recursive backtracking, tracking the current prefix and the counts of each digit used.
- At each position,... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A hospital has 5 internists and 6 surgeons. They need to dispatch 4 doctors to join a disaster relief medical team. If both internists and surgeons are required to be part of the team... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"310"
]
}
} | {
"index": 9,
"source_id": "numinamath-40",
"split": "train"
} | A hospital has 5 internists and 6 surgeons. They need to dispatch 4 doctors to join a disaster relief medical team. If both internists and surgeons are required to be part of the team, how many ways are there to choose the doctors (answer with a number)? | - Recognize the team size is 4 and that the doctors come from two distinct groups with fixed totals, with the condition that each group must contribute at least one doctor.
- Translate the condition into feasible positive count pairs for internists and surgeons that add up to the team size, while respecting the availab... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A chessboard of size $8 \\times 8$ is considered.\n\nHow many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?\n\nNow it's your turn to respo... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"564480"
]
}
} | {
"index": 10,
"source_id": "numinamath-42",
"split": "train"
} | A chessboard of size $8 \times 8$ is considered.
How many ways are there to place 6 rooks such that no two rooks are ever on the same row or column? | - Interpret the condition as requiring the six rooks to occupy six distinct rows and six distinct columns, so the first task is to select which rows and columns will be used.
- Count independently the choices of six rows from the eight available and six columns from the eight available.
- Once the row set and column se... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: The base of the pyramid is an isosceles triangle \\( A B C \\) with a base \\( AC = 2 \\) and a lateral side \\( \\sqrt{7} \\). The face \\( A C D \\) is perpendicular to the base pla... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"3"
]
}
} | {
"index": 11,
"source_id": "numinamath-43",
"split": "train"
} | The base of the pyramid is an isosceles triangle \( A B C \) with a base \( AC = 2 \) and a lateral side \( \sqrt{7} \). The face \( A C D \) is perpendicular to the base plane and is an equilateral triangle. Find the edge \( B D \), as well as the areas of all those cross-sections of the pyramid that are squares. | - Choose a coordinate system with the base plane as z=0, placing A and C along one axis so the midpoint of AC is easy to identify; use the isosceles condition to put B on the perpendicular bisector of AC.
- Determine B's position in the base plane by using the given equal lateral side lengths to solve for its height ab... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equati... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"\\frac{2}{5}"
]
}
} | {
"index": 12,
"source_id": "numinamath-50",
"split": "train"
} | I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction. | - Rearrange the equation to collect and factor the q terms, obtaining an expression for q in terms of p.
- For an integer q to exist, the resulting fraction must evaluate to an integer, which translates into a divisibility condition on p.
- Identify and handle separately the value of p that makes the denominator zero b... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let \\( a = \\frac{\\sqrt{5} + \\sqrt{3}}{\\sqrt{5} - \\sqrt{3}} \\) and \\( b = \\frac{\\sqrt{5} - \\sqrt{3}}{\\sqrt{5} + \\sqrt{3}} \\). Find the value of \\( a^4 + b^4 + (a+b)^4 \\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7938"
]
}
} | {
"index": 13,
"source_id": "numinamath-52",
"split": "train"
} | Let \( a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \) and \( b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \). Find the value of \( a^4 + b^4 + (a+b)^4 \). | - Recognize that the two given fractions are reciprocal to one another, so they have a symmetric relationship that can simplify later powers.
- Rationalize each denominator by multiplying numerator and denominator by its conjugate, then simplify using the difference of squares to remove radicals from the denominator.
-... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: The numbers \\(1, 2, \\cdots, 9\\) are randomly placed into a \\(3 \\times 3\\) grid such that each cell contains exactly one number and each number is used exactly once. What is the ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"\\frac{1}{14}"
]
}
} | {
"index": 14,
"source_id": "numinamath-53",
"split": "train"
} | The numbers \(1, 2, \cdots, 9\) are randomly placed into a \(3 \times 3\) grid such that each cell contains exactly one number and each number is used exactly once. What is the probability that the sum of the numbers in each row and each column is odd?
(A) \(\frac{1}{21}\)
(B) \(\frac{1}{14}\)
(C) \(\frac{5}{63}\)
... | - Reduce the problem to parity: a row or column has odd sum exactly when it contains an odd number of odd entries.
- Count the available odd and even entries among 1 through 9, then determine which 3×3 parity patterns can make every row and every column have an odd number of odd entries.
- Characterize those valid pari... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let \\( B \\) be a point on the circle centered at \\( O \\) with diameter \\( AC \\) and let \\( D \\) and \\( E \\) be the circumcenters of the triangles \\( OAB \\) and \\( OBC \\)... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"45"
]
}
} | {
"index": 15,
"source_id": "numinamath-55",
"split": "train"
} | Let \( B \) be a point on the circle centered at \( O \) with diameter \( AC \) and let \( D \) and \( E \) be the circumcenters of the triangles \( OAB \) and \( OBC \) respectively. Given that \( \sin \angle BOC = \frac{4}{5} \) and \( AC = 24 \), find the area of the triangle \( BDE \). | - Since AC is a diameter, treat O as the midpoint of AC and place O at the origin with AC along a coordinate axis, so the circle radius is determined from the diameter.
- Use the given sine of angle BOC, the circle radius, and the Pythagorean identity to assign coordinates to B on the circle; choose a convenient orient... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A positive integer will be called \"sparkly\" if its smallest positive divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers \\(2, 3, \\ldo... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"3"
]
}
} | {
"index": 16,
"source_id": "numinamath-59",
"split": "train"
} | A positive integer will be called "sparkly" if its smallest positive divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers \(2, 3, \ldots, 2003\) are sparkly? | - Translate the definition of "sparkly" into two computable quantities for a candidate n: its smallest positive divisor greater than 1, and its total number of positive divisors including 1.
- Treat the task as a finite enumeration over the integers from 2 through 2003, since the condition can be checked directly for e... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given that \\( x \\geq 1 \\), \\( y \\geq 1 \\), and \\(\\lg ^{2} x+\\lg ^{2} y=\\lg 10 x^{2}+\\lg 10 y^{2}\\), find the maximum value of \\( u=\\lg x y \\).\n\nNow it's your turn to ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"4.82842712474619"
]
}
} | {
"index": 17,
"source_id": "numinamath-61",
"split": "train"
} | Given that \( x \geq 1 \), \( y \geq 1 \), and \(\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}\), find the maximum value of \( u=\lg x y \). | - Introduce logarithmic variables, such as setting a = lg x and b = lg y, and note that the constraints x ≥ 1 and y ≥ 1 imply a ≥ 0 and b ≥ 0.
- Simplify the right-hand side using logarithm rules for products and powers so the given equation is expressed entirely in terms of a and b.
- Rearrange the resulting relation ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Calculate the lengths of the arcs of the curves given by the parametric equations.\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array}{l}\nx=\\left(t^{2}-2\\right) \\sin t+2 t \\cos t ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"82.683"
]
}
} | {
"index": 18,
"source_id": "numinamath-72",
"split": "train"
} | Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$ | - Recognize that the length of a parametric curve is found by integrating its speed, sqrt((dx/dt)^2 + (dy/dt)^2), over the given parameter interval.
- Differentiate both x(t) and y(t) with respect to t, using the product rule and grouping similar terms.
- Before integrating, simplify the derivatives and the sum of thei... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Calculate the limit of the function:\n\n\\[\n\\lim _{x \\rightarrow 0} \\frac{\\cos x+\\ln (1+x) \\sqrt{2+\\cos \\left(\\frac{1}{x}\\right)}}{2+e^{x}}\n\\]\n\nNow it's your turn to re... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"\\frac{1}{3}"
]
}
} | {
"index": 19,
"source_id": "numinamath-77",
"split": "train"
} | Calculate the limit of the function:
\[
\lim _{x \rightarrow 0} \frac{\cos x+\ln (1+x) \sqrt{2+\cos \left(\frac{1}{x}\right)}}{2+e^{x}}
\] | - First check the domain near x=0: the logarithm requires 1+x>0, and the square-root argument is nonnegative because cosine is bounded.
- Separate the numerator into a well-behaved term and an oscillatory term; recognize that cos x has a finite limit as x→0.
- For the term ln(1+x)√(2+cos(1/x)), observe that ln(1+x) ten... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A Louis Vuitton store in Shanghai had a number of pairs of sunglasses which cost an average of \\ $$ 900 $ per pair. LiAngelo Ball stole a pair which cost \\$ $ 2000$ . Afterwards, ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"111"
]
}
} | {
"index": 20,
"source_id": "numinamath-79",
"split": "train"
} | A Louis Vuitton store in Shanghai had a number of pairs of sunglasses which cost an average of \ $$ 900 $ per pair. LiAngelo Ball stole a pair which cost \$ $ 2000$ . Afterwards, the average cost of sunglasses in the store dropped to \ $$ 890$ per pair. How many pairs of sunglasses were in the store before LiAngelo B... | - Define n as the original number of pairs and use the average-cost relationship to express the original total value in terms of n.
- Recognize that after one pair is removed, the remaining count is n-1, so the new average determines the remaining total value.
- Form an equation by equating that remaining total with th... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?\n(a) 778152\n(b) 781569\n(... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"879103"
]
}
} | {
"index": 21,
"source_id": "numinamath-84",
"split": "train"
} | What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?
(a) 778152
(b) 781569
(c) 879103
(d) 986103
(e) 987776 | - Recognize that the task is to select an ordered subsequence of six digits from the given number, deleting the rest while preserving relative order.
- Proceed greedily from left to right: for each digit of the result, scan a window starting at the current position whose size extends only as far as the number of deleti... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: To enhance students' mathematical literacy, the school offers three elective courses for 11th-grade students: \"History of Mathematics\", \"Symmetry and Groups\", and \"Geometry on th... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"12"
]
}
} | {
"index": 22,
"source_id": "numinamath-90",
"split": "train"
} | To enhance students' mathematical literacy, the school offers three elective courses for 11th-grade students: "History of Mathematics", "Symmetry and Groups", and "Geometry on the Sphere". It is known that there are 600 students in the 11th grade, and each of them participates in only one elective course. To understand... | - Recognize that the three course enrollments form an arithmetic sequence; choose the History of Mathematics enrollment as the first term and express the other two using the given common difference.
- Use the fact that the three enrollment counts sum to the total number of 11th-grade students to determine the History e... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given that point $F$ is the focus of the parabola $C$: $y^{2}=4x$, and point $B$ is on parabola $C$, $A(5,4)$, when the perimeter of $\\triangle ABF$ is minimum, the area of this tria... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2"
]
}
} | {
"index": 23,
"source_id": "numinamath-91",
"split": "train"
} | Given that point $F$ is the focus of the parabola $C$: $y^{2}=4x$, and point $B$ is on parabola $C$, $A(5,4)$, when the perimeter of $\triangle ABF$ is minimum, the area of this triangle is $\_\_\_\_\_\_$. | - Identify the focus of the parabola from its standard form, and note that the distance from A to F is fixed, so the perimeter is minimized by minimizing only the variable sum of the two remaining side lengths.
- Represent a general point B on the parabola with a single parameter using its standard parameterization, so... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Solve\n\\[\\arcsin x + \\arcsin 2x = \\frac{\\pi}{3}.\\]\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This process MUST be enc... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"0.3273"
]
}
} | {
"index": 24,
"source_id": "numinamath-92",
"split": "train"
} | Solve
\[\arcsin x + \arcsin 2x = \frac{\pi}{3}.\] | - Determine the permissible values of \(x\) by imposing the arcsine domain on both \(\arcsin x\) and \(\arcsin 2x\), then intersect those intervals to obtain the valid search range.
- Rewrite the equation as a function of \(x\) whose zero corresponds to the original equality, and note that this function is continuous o... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given the equation $5x^2+kx-10=0$ has one root as $-5$, find the other root and the value of $k$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a re... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"23"
]
}
} | {
"index": 25,
"source_id": "numinamath-93",
"split": "train"
} | Given the equation $5x^2+kx-10=0$ has one root as $-5$, find the other root and the value of $k$. | - Recognize that the given root and the unknown other root are the two roots of the quadratic; denote the unknown one by a variable.
- Use Vieta's formulas for a quadratic, relating the product of the roots to the constant term divided by the leading coefficient and the sum of the roots to the negative middle coefficie... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let \\( S = \\{1, 2, \\cdots, n\\} \\). Suppose the 3-element subsets \\( A_1, A_2, \\cdots, A_m \\) (\\( m > 1 \\)) satisfy:\n(1) Every 2-element subset of \\( S \\) is contained in ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7"
]
}
} | {
"index": 26,
"source_id": "numinamath-95",
"split": "train"
} | Let \( S = \{1, 2, \cdots, n\} \). Suppose the 3-element subsets \( A_1, A_2, \cdots, A_m \) (\( m > 1 \)) satisfy:
(1) Every 2-element subset of \( S \) is contained in exactly one \( A_i \) (\( 1 \leq i \leq m \));
(2) Each \( A_i \) and \( A_j \) (\( i \neq j, 1 \leq i, j \leq m \)) have exactly one common element.
... | - Recognize that the conditions describe a Steiner triple system: every pair of elements lies in exactly one triple, so count all 2-element subsets by triples to relate m and n.
- Use the pairing condition to determine how many triples contain a fixed element, since that element must be paired once with each other elem... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given that the function $f(x)=\\sin (π-ωx)\\cos ωx+\\cos ^{2}ωx\\ (ω > 0)$ has a minimum positive period of $π$.\n(I) Find the value of $ω$;\n(II) The horizontal coordinates of each p... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"1"
]
}
} | {
"index": 27,
"source_id": "numinamath-100",
"split": "train"
} | Given that the function $f(x)=\sin (π-ωx)\cos ωx+\cos ^{2}ωx\ (ω > 0)$ has a minimum positive period of $π$.
(I) Find the value of $ω$;
(II) The horizontal coordinates of each point on the graph of the function $y=f(x)$ are shortened to half of the original, and the vertical coordinates remain unchanged, resulting in t... | - Simplify f using identities such as sin(π−θ)=sin θ and appropriate square/double-angle identities, so that f is expressed in terms of ωx.
- Use the given minimum positive period to determine ω>0 by comparing it with the periods of the simplified trigonometric components, and verify that no smaller common period remai... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.\n\nAnton drove the car for half the ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"0.4"
]
}
} | {
"index": 28,
"source_id": "numinamath-103",
"split": "train"
} | Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.
Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What frac... | - Since the entire trip is at constant speed, recognize that distance traveled is directly proportional to time, so the requested distance fraction equals Vasya’s driving time divided by total time.
- Introduce variables for each person’s driving time and the total time, then translate every verbal condition into an eq... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In the sequence $\\{a\\_n\\}$, $a\\_n=10^{ \\frac {n}{11}}$, let $T\\_n=a\\_1 \\cdot a\\_2 \\cdot \\ldots \\cdot a\\_n$. Find the smallest positive integer $n$ such that $T\\_n > 10^{... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"11"
]
}
} | {
"index": 29,
"source_id": "numinamath-104",
"split": "train"
} | In the sequence $\{a\_n\}$, $a\_n=10^{ \frac {n}{11}}$, let $T\_n=a\_1 \cdot a\_2 \cdot \ldots \cdot a\_n$. Find the smallest positive integer $n$ such that $T\_n > 10^{5}$. | - Recognize that each term is a power of 10 whose exponent depends linearly on the index, so the product can be rewritten as a single power of 10 with the exponents added.
- Use the arithmetic-series formula to sum those exponents, expressing T_n as 10 raised to a quadratic expression in n.
- Compare T_n with 10^5; bec... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: If the distance from point P on the hyperbola $\\frac{x^2}{9} - \\frac{y^2}{7} = 1$ to the right focus is 1, then the distance from point P to the origin is _______.\n\nNow it's your ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"3"
]
}
} | {
"index": 30,
"source_id": "numinamath-113",
"split": "train"
} | If the distance from point P on the hyperbola $\frac{x^2}{9} - \frac{y^2}{7} = 1$ to the right focus is 1, then the distance from point P to the origin is _______. | - Recognize the equation as a standard hyperbola centered at the origin with its transverse axis along the x-axis; locate the right focus using the focal relation from the coefficients.
- Let P=(x,y). Translate the given distance from P to the right focus into a circle equation centered at that focus.
- Combine this ci... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let $S = 1 + \\frac{1}{\\sqrt2}+ \\frac{1}{\\sqrt3}+\\frac{1}{\\sqrt4}+...+ \\frac{1}{\\sqrt{99}}+ \\frac{1}{\\sqrt{100}}$ . Find $[S]$ .\n\nYou may use the fact that $\\sqrt{n} <... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"18"
]
}
} | {
"index": 31,
"source_id": "numinamath-115",
"split": "train"
} | Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$ .
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$ . | - Recognize that S is a finite sum of reciprocal square-root terms over a fixed integer range, so the problem asks for an accurate total before extracting its integer part.
- Choose direct numerical summation over that entire range instead of trying to find an exact closed-form simplification for the finite series.
- A... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Simplify first, then evaluate: $-(m^{2}-3m)+2(m^{2}-m-1)$, where $m=-1$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This pro... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"-2"
]
}
} | {
"index": 32,
"source_id": "numinamath-118",
"split": "train"
} | Simplify first, then evaluate: $-(m^{2}-3m)+2(m^{2}-m-1)$, where $m=-1$. | - Begin by distributing the minus sign across the first parentheses and the factor 2 across the second parentheses, carefully tracking how each sign changes.
- Combine like terms by grouping the \(m^2\) terms, the \(m\) terms, and the constant terms into one simplified polynomial.
- Substitute \(m=-1\) into the simplif... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A natural number \\( x = 5^n - 1 \\) is given, where \\( n \\) is a natural number. It is known that \\( x \\) has exactly three distinct prime factors, one of which is 11. Find \\( x... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"3124"
]
}
} | {
"index": 33,
"source_id": "numinamath-119",
"split": "train"
} | A natural number \( x = 5^n - 1 \) is given, where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime factors, one of which is 11. Find \( x \). | - Treat the exponent n as the unknown parameter: for any natural n, the candidate value is determined by x = 5^n - 1.
- Use a systematic enumeration over natural n, starting from the smallest possible exponents and increasing one at a time.
- For each candidate n, compute 5^n - 1 exactly and factor it, listing only its... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{9}{2}\\quad \\t... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"6"
]
}
} | {
"index": 34,
"source_id": "numinamath-121",
"split": "train"
} | Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC.$ if $OQ$ is 3 inches, then $OP$, in inches, is:
$\text{(A) } 3\quad \text{(B) } \frac{9}{2}\quad \text{(C) } 6\quad \text{(D) } 9\quad \text{(E) } \text{undetermined}$ | - Recognize O as the centroid, the common intersection of the medians, so it lies on both AP and CQ and P, Q are side midpoints.
- Recall the centroid theorem: each median is divided in a 2:1 ratio, with the longer portion adjacent to the vertex.
- Identify OQ as the shorter portion of median CQ and use it to establish... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxe... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"28182"
]
}
} | {
"index": 35,
"source_id": "numinamath-124",
"split": "train"
} | There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes wer... | - Track two quantities separately: the total number of boxes and the number of non-empty boxes. Each operation inserts several new nested boxes into one recipient box, so the recipient becomes non-empty while the total box count increases.
- To reach the required number of non-empty boxes, determine how many operations... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: There are 211 students and four different types of chocolates, with more than 633 pieces of each type of chocolate. Each student can take up to three pieces of chocolate, or none at a... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7"
]
}
} | {
"index": 36,
"source_id": "numinamath-126",
"split": "train"
} | There are 211 students and four different types of chocolates, with more than 633 pieces of each type of chocolate. Each student can take up to three pieces of chocolate, or none at all. Grouping students based on both the type and number of chocolates they take, the maximum number of students in any group is at least ... | - Interpret each student’s choice as an entire chocolate-taking pattern: how many pieces of each of the four types are taken, not just the total number of pieces.
- Enumerate all feasible patterns for 0, 1, 2, or 3 pieces, grouping together choices with the same multiset of types; the large supply of each type ensures ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A carpenter took a wooden square and cut 4 smaller equal squares from it, each with an area of $9\\%$ of the area of the large square. The remaining area of the original square was $2... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"20"
]
}
} | {
"index": 37,
"source_id": "numinamath-129",
"split": "train"
} | A carpenter took a wooden square and cut 4 smaller equal squares from it, each with an area of $9\%$ of the area of the large square. The remaining area of the original square was $256 \mathrm{~cm}^{2}$. Find the side length of the original square in centimeters. | - Introduce a variable for the original square’s side length, and represent the original area in terms of that variable.
- Recognize that each removed smaller square’s area is a percentage of the original area, then combine the four equal removed areas into one total removed fraction.
- Use the given remaining area to ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimmin... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"11"
]
}
} | {
"index": 38,
"source_id": "numinamath-138",
"split": "train"
} | In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visi... | - Treat the sauna admission as the unknown, and use the fact that the old combined pass price is the sum of the sauna cost and the pool-only cost.
- Incorporate the condition that the pool-only price exceeds the sauna price by the stated amount, forming an equation that determines the sauna cost and then the pool-only ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: There is a five-digit positive odd number \\( x \\). In \\( x \\), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a ne... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"29995"
]
}
} | {
"index": 39,
"source_id": "numinamath-140",
"split": "train"
} | There is a five-digit positive odd number \( x \). In \( x \), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x + 1) \), what is the value of \( x \)?
(Chinese Junior Hi... | - Identify that x is a five-digit odd number, so restrict attention to the finite set of five-digit integers ending in an odd digit and begin at the smallest such value.
- Treat the replacement rule as a digit-wise map: each digit 2 becomes 5, each 5 becomes 2, and all other digits are unchanged, preserving place value... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given positive integers $a$, $b$, $c$, and $d$ satisfying $a < b < c < d$ and $ab + bc + ac = abc = d$, find the value of $d$.\n\nNow it's your turn to respond for the current step.\n... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"36"
]
}
} | {
"index": 40,
"source_id": "numinamath-146",
"split": "train"
} | Given positive integers $a$, $b$, $c$, and $d$ satisfying $a < b < c < d$ and $ab + bc + ac = abc = d$, find the value of $d$. | - Recognize that both relations describe the same quantity d, so eliminate d by setting the pairwise-sum expression equal to the product expression involving a, b, and c.
- Use the positivity and strict ordering a < b < c to consider only increasing triples, avoiding duplicates and limiting the search.
- Systematically... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"45"
]
}
} | {
"index": 41,
"source_id": "numinamath-157",
"split": "train"
} | In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong or unanswered earns 0 points. After grading, the teacher found that there were students with each possible score, and for each score, th... | - Interpret each student’s result as an ordered pair: the score earned on the first question and the score earned on the second question, since the two questions are distinct.
- Recognize that each question can take one of three possible outcomes, corresponding to the allowed point values, so the score distributions ar... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: The number $\\text{N}$ is between $9$ and $17$. The [average](https://artofproblemsolving.com/wiki/index.php/Average) of $6$, $10$, and $\\text{N}$ could be\n$\\text{(A)}\\ 8 \\qquad... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"10"
]
}
} | {
"index": 42,
"source_id": "numinamath-160",
"split": "train"
} | The number $\text{N}$ is between $9$ and $17$. The [average](https://artofproblemsolving.com/wiki/index.php/Average) of $6$, $10$, and $\text{N}$ could be
$\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$ | - Interpret the condition: the average of the three quantities 6, 10, and N must equal one of the listed candidate values.
- Use the definition of average to form an equation relating N to a candidate average, then rearrange it so N is expressed in terms of that candidate average.
- For each candidate average, check wh... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: \\(ABCD\\) is a cyclic quadrilateral in which \\(AB = 3\\), \\(BC = 5\\), \\(CD = 6\\), and \\(AD = 10\\). \\(M\\), \\(I\\), and \\(T\\) are the feet of the perpendiculars from \\(D\\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2.7777777777777777"
]
}
} | {
"index": 43,
"source_id": "numinamath-168",
"split": "train"
} | \(ABCD\) is a cyclic quadrilateral in which \(AB = 3\), \(BC = 5\), \(CD = 6\), and \(AD = 10\). \(M\), \(I\), and \(T\) are the feet of the perpendiculars from \(D\) to the lines \(AB\), \(AC\), and \(BC\) respectively. Determine the value of \(MI / IT\). | - Interpret M and I as the perpendicular projections of D onto the two lines through A, AB and AC; similarly, I and T are projections of D onto the two lines through C, AC and BC.
- For two lines meeting at a vertex, relate the separation of the two projection feet to the segment from that vertex to D and the sine of t... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A bag contains 60 colored balls, with 6 balls of each color. Blindly drawing balls from the bag, in order to ensure that there are three balls of the same color among the drawn balls,... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"21"
]
}
} | {
"index": 44,
"source_id": "numinamath-177",
"split": "train"
} | A bag contains 60 colored balls, with 6 balls of each color. Blindly drawing balls from the bag, in order to ensure that there are three balls of the same color among the drawn balls, at least 21 balls need to be drawn. | - Identify how many distinct colors are present from the total number of balls and the equal number of balls per color, since the condition concerns a color reaching multiplicity three.
- Because the draw is blind, reason about the worst-case ordering that delays any color appearing three times for as long as possible.... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In order to understand the academic level of students in a certain high school, the education department conducted stratified sampling by grade, selecting 100 students from a total of... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"35"
]
}
} | {
"index": 45,
"source_id": "numinamath-184",
"split": "train"
} | In order to understand the academic level of students in a certain high school, the education department conducted stratified sampling by grade, selecting 100 students from a total of 2400 students in the school. If there are 840 students in the first grade of the school, then the number of first-grade students that sh... | - Recognize that stratified sampling by grade requires the sample to preserve each grade’s proportion from the whole school.
- Find the first-grade proportion by comparing the number of first-grade students with the total school enrollment.
- Apply that same proportion to the total sample size to determine the number o... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other.\n\nNow it's your turn to respond fo... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7744"
]
}
} | {
"index": 46,
"source_id": "numinamath-188",
"split": "train"
} | Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other. | - Interpret the unknown as a four-digit number with digit pattern ABBA, so the first two digits match and the last two digits match, with the leading digit nonzero.
- Restrict attention to the four-digit range, since any candidate must lie between the smallest and largest four-digit values.
- For each candidate, test t... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let $p$ be a prime. It is given that there exists a unique nonconstant function $\\chi:\\{1,2,\\ldots, p-1\\}\\to\\{-1,1\\}$ such that $\\chi(1) = 1$ and $\\chi(mn) = \\chi(m)... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"24"
]
}
} | {
"index": 47,
"source_id": "numinamath-189",
"split": "train"
} | Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$ ). For how many positive primes $p$ less than $100$ ... | - Interpret χ as a multiplicative character on the cyclic group of nonzero residues modulo p; for odd p, the unique nonconstant such character is the Legendre symbol, classifying each a as a quadratic residue or nonresidue.
- Handle the prime p=2 separately, since its nonzero residue domain has only one element and the... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: $C_{99}^{98} + C_{99}^{97} =$ （Answer in digits）\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This process MUST be enclosed wi... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"4950"
]
}
} | {
"index": 48,
"source_id": "numinamath-191",
"split": "train"
} | $C_{99}^{98} + C_{99}^{97} =$ （Answer in digits） | - Recognize both terms as binomial coefficients with the same upper index and consecutive lower indices.
- Recall the identity that combines adjacent binomial coefficients, namely Pascal's rule relating terms with lower indices differing by one.
- Match the given lower indices to that identity, then rewrite the sum as ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Calculate:<br/>$(1)-{3}^{3}×(-\\frac{1}{3})+|-2|÷{(-\\frac{1}{2})}^{2}$;<br/>$(2)7-12×(\\frac{2}{3}-\\frac{3}{4}+\\frac{5}{6})$.\n\nNow it's your turn to respond for the current step.... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"-2"
]
}
} | {
"index": 49,
"source_id": "numinamath-194",
"split": "train"
} | Calculate:<br/>$(1)-{3}^{3}×(-\frac{1}{3})+|-2|÷{(-\frac{1}{2})}^{2}$;<br/>$(2)7-12×(\frac{2}{3}-\frac{3}{4}+\frac{5}{6})$. | - Treat the two calculations independently, and for each apply the order of operations: powers and absolute values first, then multiplication/division, then addition/subtraction.
- In the first expression, pay attention to the negative sign attached to the power and to the subtraction of a negative factor; evaluate the... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Fill a $3 \\times 3$ grid with three 1s, three 2s, and three 3s, ensuring that each row and column contains no repeated numbers. How many different ways are there to fill the grid?\n\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"12"
]
}
} | {
"index": 50,
"source_id": "numinamath-195",
"split": "train"
} | Fill a $3 \times 3$ grid with three 1s, three 2s, and three 3s, ensuring that each row and column contains no repeated numbers. How many different ways are there to fill the grid? | - Interpret the filling as a 3x3 Latin-square-type arrangement: since each row and column has length three and must avoid repeats, every row and every column must be a permutation of the three symbols.
- Represent a candidate grid as an ordering of the multiset containing three copies of each symbol, and enumerate dist... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A man travels 10 km at a speed of 4 km/h and another 10 km at a speed of 6 km/h. If the average speed of the whole journey is \\( x \\) km/h, find \\( x \\).\n\nNow it's your turn to ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"4.8"
]
}
} | {
"index": 51,
"source_id": "numinamath-197",
"split": "train"
} | A man travels 10 km at a speed of 4 km/h and another 10 km at a speed of 6 km/h. If the average speed of the whole journey is \( x \) km/h, find \( x \). | - Recognize that average speed for a multi-leg journey means total distance divided by total elapsed time, not the arithmetic mean of the two speeds.
- For each leg, use the relationship between speed, distance, and time to determine the time spent on that leg from its given distance and speed.
- Add the two leg times ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let $\\{a_i\\}_{i=0}^\\infty$ be a sequence of real numbers such that \\[\\sum_{n=1}^\\infty\\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\\cdots\\] for all $|x|<1$ . Find $a_{1000... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"16"
]
}
} | {
"index": 52,
"source_id": "numinamath-198",
"split": "train"
} | Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$ . Find $a_{1000}$ .
*Proposed by David Altizio* | - Expand each summand as a geometric series in powers of x^n; this is valid because |x|<1 and each x^n lies in the unit disk.
- Treat the original expression as a double series over n and the exponent multiple, then interchange summation, justified by absolute convergence.
- Fix a power x^k and count how many summands ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.\n2. If $a+3b=3$, find the value of $3^{a}\\times 27^{b}$.\n3. Given $8\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"-1"
]
}
} | {
"index": 53,
"source_id": "numinamath-200",
"split": "train"
} | Answer the following questions as required.
1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.
2. If $a+3b=3$, find the value of $3^{a}\times 27^{b}$.
3. Given $8\times 2^{m}\div 16^{m}=2^{6}$, find the value of $m$. | - In the first part, interpret 10^(m-n) as a quotient of the two given powers of 10, then use the quotient-of-powers rule to rewrite it in terms of the known values.
- In the second part, express 27 as a power of 3, then combine the factors using exponent rules so the whole expression becomes a single power of 3 with e... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{200... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"46"
]
}
} | {
"index": 54,
"source_id": "numinamath-201",
"split": "train"
} | Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers (they can be the same) such that \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different from each other. What is the minimum number of distinct numbers in \( a_{1}, a_{2}, \cdots, a_{2006} \)? | - Recognize that the condition concerns consecutive quotients, so the task is to find the smallest palette of distinct positive integer values that can support all required distinct adjacent ratios.
- If the sequence uses n distinct values, count the distinct quotients available from ordered pairs of those values; equa... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: For what smallest natural number \\( a \\) are there exactly 50 perfect squares in the numerical interval \\( (a, 3a) \\)?\n\nNow it's your turn to respond for the current step.\nYou ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"4486"
]
}
} | {
"index": 55,
"source_id": "numinamath-203",
"split": "train"
} | For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)? | - Recognize that perfect squares are numbers of the form n^2 for integer n, so focus on which integers n satisfy the strict interval condition for a candidate a.
- Convert the inequality a < n^2 < 3a into bounds involving square roots, yielding a condition on n between sqrt(a) and sqrt(3a).
- Translate the strict endpo... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: From the numbers 1, 2, 3, and 4, form a four-digit number where the first digit is 1 and only two digits are the same. There are $\\boxed{\\_\\_\\_\\_}$ such four-digit numbers.\n\nNo... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"18"
]
}
} | {
"index": 56,
"source_id": "numinamath-214",
"split": "train"
} | From the numbers 1, 2, 3, and 4, form a four-digit number where the first digit is 1 and only two digits are the same. There are $\boxed{\_\_\_\_}$ such four-digit numbers. | - Fix the leading digit as 1, then focus on filling the next three positions using the remaining available digits without allowing an unwanted extra repeat of 1.
- Since exactly one pair of digits repeats in the whole number, treat the last three positions as a multiset containing one repeated digit and one distinct si... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Oleg has 550 rubles and wants to give his mother tulips for March 8th. The number of tulips must be odd, and each tulip must be of a different shade. In the store, each tulip costs 49... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"1024"
]
}
} | {
"index": 57,
"source_id": "numinamath-221",
"split": "train"
} | Oleg has 550 rubles and wants to give his mother tulips for March 8th. The number of tulips must be odd, and each tulip must be of a different shade. In the store, each tulip costs 49 rubles, and there are eleven different shades available. How many ways are there for Oleg to give the flowers to his mother? Provide the... | - Interpret a purchase as choosing a subset of the distinct tulip shades: the subset size is the number of tulips, and the total cost is that size times the fixed price per tulip.
- Use the budget to determine the largest possible subset size, then compare it with the number of available shades to identify exactly whic... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.\n\n<details><summary>Note</summary>Note. Here (and generally in MathLin... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7960"
]
}
} | {
"index": 58,
"source_id": "numinamath-222",
"split": "train"
} | Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
<details><summary>Note</summary>Note. Here (and generally in MathLinks) natural numbers supposed to be positive.</details> | - Identify that trailing zeros in a factorial come from factors of 10, so they are limited by the number of factors of 5 rather than 2 in the product 1·2·...·n.
- Express the number of trailing zeros Z(n) as the exponent of 5 in n!, using Legendre's formula: sum of floor(n/5^k) over positive k.
- Since Z(n) is nondecre... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.\n\n*2018 CCA Math Bonanza Lightning Round #3.1*\n\nNow it's your turn to ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"264"
]
}
} | {
"index": 59,
"source_id": "numinamath-224",
"split": "train"
} | The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
*2018 CCA Math Bonanza Lightning Round #3.1* | - Recognize that the expression has the form \(a^4+a^2+1\) with \(a=16\), which suggests using a factorization identity rather than expanding directly.
- Apply the standard factorization for this form to split the number into two smaller factors, each more amenable to prime factorization.
- Factor each resulting factor... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Let the sequence \\\\(\\{a_n\\}\\) be defined by \\\\(a_n = 2n - 7 (n \\in \\mathbb{N}^*)\\), then the value of \\\\(|a_1| + |a_2| + \\ldots + |a_{15}| = \\) ______.\n\nNow it's your ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"153"
]
}
} | {
"index": 60,
"source_id": "numinamath-228",
"split": "train"
} | Let the sequence \\(\{a_n\}\) be defined by \\(a_n = 2n - 7 (n \in \mathbb{N}^*)\), then the value of \\(|a_1| + |a_2| + \ldots + |a_{15}| = \) ______. | - Recognize that the sequence is arithmetic with a positive slope, so its terms change sign at most once; determine the cutoff index by comparing when a_n is negative versus nonnegative.
- Split the required sum over n=1 to 15 into two parts using that cutoff, removing absolute values by negating the negative terms and... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same g... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"27720"
]
}
} | {
"index": 61,
"source_id": "numinamath-235",
"split": "train"
} | On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process conti... | - Recognize that a row size is possible exactly when it divides the total number of students; the June 13 condition means the total has exactly twelve positive divisors.
- Include the given row sizes as required divisors, so the total must be divisible by their least common multiple.
- Use the divisor-count formula: if... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Yura left home for school 5 minutes later than Lena, but he walked twice as fast as her. How long after Yura started walking will he catch up to Lena?\n\nNow it's your turn to respond... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"5"
]
}
} | {
"index": 62,
"source_id": "numinamath-237",
"split": "train"
} | Yura left home for school 5 minutes later than Lena, but he walked twice as fast as her. How long after Yura started walking will he catch up to Lena? | - Let the unknown be the time from Yura's start until the catch-up, and introduce a common speed scale for Lena; Yura's speed is twice that scale.
- Because they meet, both have traveled the same distance from home; Lena's travel time includes her earlier head start, while Yura's travel time is just the unknown.
- Expr... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find the greatest common divisor of $5616$ and $11609$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This process MUST be encl... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"13"
]
}
} | {
"index": 63,
"source_id": "numinamath-239",
"split": "train"
} | Find the greatest common divisor of $5616$ and $11609$. | - Recognize that the task is to compute the greatest common divisor of two integers, so a method that reduces the pair while preserving the GCD is appropriate.
- Use the Euclidean algorithm: repeatedly divide the larger current value by the smaller and keep the remainder, treating this remainder as the new smaller term... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: From 22 kg of fresh mushrooms, 2.5 kg of dried mushrooms are obtained, which still contain 12% water. What percent of water is in the fresh mushrooms?\n\nNow it's your turn to respond... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"90\\%"
]
}
} | {
"index": 64,
"source_id": "numinamath-240",
"split": "train"
} | From 22 kg of fresh mushrooms, 2.5 kg of dried mushrooms are obtained, which still contain 12% water. What percent of water is in the fresh mushrooms? | - Recognize that drying removes water while the non-water dry matter remains unchanged, so conservation of dry matter links the fresh and dried mushrooms.
- Use the stated water percentage in the dried mushrooms to split the dried mass into water and dry matter.
- Treat that dry matter as the same dry matter present in... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: It is known that among the members of the government of Limonia (a total of 20 members), there is at least one honest member, and among any two members, at least one is a bribe-taker.... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"19"
]
}
} | {
"index": 65,
"source_id": "numinamath-244",
"split": "train"
} | It is known that among the members of the government of Limonia (a total of 20 members), there is at least one honest member, and among any two members, at least one is a bribe-taker. How many bribe-takers are in the government? | - Classify members into honest and bribe-takers, and interpret the condition as applying to every possible pair of members.
- From “among any two members, at least one is a bribe-taker,” infer that two honest members cannot coexist, so the number of honest members cannot exceed one.
- Combine this upper bound with the ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In a drama club, the number of boys is $80 \\%$ of the number of girls. What percentage is the number of girls in this drama club relative to the number of boys?\n\nNow it's your turn... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"125\\%"
]
}
} | {
"index": 66,
"source_id": "numinamath-248",
"split": "train"
} | In a drama club, the number of boys is $80 \%$ of the number of girls. What percentage is the number of girls in this drama club relative to the number of boys? | - Identify the two unknown quantities being compared: the number of boys and the number of girls, and assign symbols to represent them.
- Translate the phrase “boys is 80% of girls” into a relationship between the two quantities, expressing the boy count in terms of the girl count.
- Recognize that “girls relative to b... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A particle moves according to the law $s=2t^3$. The average speed of the particle in the time interval $[1, 1.1]$ is \\_\\_\\_\\_\\_\\_ m/s, and the instantaneous speed at $t=1$ is \\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"6"
]
}
} | {
"index": 67,
"source_id": "numinamath-252",
"split": "train"
} | A particle moves according to the law $s=2t^3$. The average speed of the particle in the time interval $[1, 1.1]$ is \_\_\_\_\_\_ m/s, and the instantaneous speed at $t=1$ is \_\_\_\_\_\_ m/s. | - Recognize that average speed over a time interval comes from the change in the position function divided by the elapsed time, so use the given position law at the interval endpoints.
- Form this as a difference quotient for the specified interval, ensuring the elapsed time is the denominator and that speed is treated... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$. A certain company can produce up to $10$ lithography machines of a certain type... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"7"
]
}
} | {
"index": 68,
"source_id": "numinamath-257",
"split": "train"
} | In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$. A certain company can produce up to $10$ lithography machines of a certain type per month. The revenue function for producing $x$ machines ($x \geqslant 1$, $x \in \mathbb{N}^{*}$) is $R(x) = x^2 + \frac{16}{x^2} + 4... | - Identify the relevant functions for each part: revenue R(x), cost C(x), marginal cost MC(x) from the definition Mf(x)=f(x+1)-f(x), and profit z(x)=R(x)-C(x).
- Restrict to the stated domain x>=1, differentiate each target function, set the derivative to zero, and solve for stationary points.
- Discard any stationary ... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning pro... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"797"
]
}
} | {
"index": 69,
"source_id": "numinamath-259",
"split": "train"
} | Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$. | - Translate “multiple of 1000” into the congruence \(2^n+5^n-n\equiv0\pmod{1000}\), and note that \(n\) must be a positive integer.
- Since the condition can be checked for any concrete candidate, treat the task as finding the smallest positive \(n\) satisfying this modular condition rather than solving a closed-form e... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: There are 27 ones written on the board. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, then eats the number of candies equal to the product of t... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"351"
]
}
} | {
"index": 70,
"source_id": "numinamath-262",
"split": "train"
} | There are 27 ones written on the board. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, then eats the number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 27 minutes? | - Interpret each minute as merging two current board numbers into their sum and adding the product of the merged pair to a running candy total.
- Note that every merge reduces the count of board numbers by one; track how many successful merges are possible before only one number remains, and do not exceed that count.
-... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: If \\( m \\) and \\( n \\) are positive integers with \\( m^2 - n^2 = 43 \\), find the value of \\( m^3 - n^3 \\).\n\nNow it's your turn to respond for the current step.\nYou should f... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"1387"
]
}
} | {
"index": 71,
"source_id": "numinamath-271",
"split": "train"
} | If \( m \) and \( n \) are positive integers with \( m^2 - n^2 = 43 \), find the value of \( m^3 - n^3 \). | - Rewrite the given relation as a difference of squares and factor it into a product of two integer expressions.
- Since the right-hand side is prime, use the fact that its positive factors are very restricted; positivity of m and n makes both factors positive and determines their relative order.
- Match the smaller fa... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: A class needs to select 4 students from 8 to participate in the 4×100 meters relay race at the school sports meeting. Among them, students A and B must be selected, and they must run ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"60"
]
}
} | {
"index": 72,
"source_id": "numinamath-275",
"split": "train"
} | A class needs to select 4 students from 8 to participate in the 4×100 meters relay race at the school sports meeting. Among them, students A and B must be selected, and they must run either the first or the last leg. The total number of different arrangements is (answer in numbers). | - Treat the relay legs as distinct ordered positions, with the first and last positions subject to a special restriction.
- Separate the task into stages: placing A and B in the two endpoint positions, choosing the other two selected students, and then ordering those two in the middle legs.
- Since A and B must both be... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: How many ordered triples of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $(a,b,c)$, with $a \\ge 2$, $b\\ge 1$, and $c \\ge 0$, satisfy both $\\log_a b = c^{2005... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2"
]
}
} | {
"index": 73,
"source_id": "numinamath-276",
"split": "train"
} | How many ordered triples of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?
$\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4$ | - Rewrite the logarithmic condition in exponential form, using that the base is at least 2 and the argument is positive, so the equation relates b directly to a and c.
- Substitute this relation into the sum constraint, then use the explosive size of the exponent c^{2005}: for c >= 2 the required b is far larger than t... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given $x + 3y = 5$ and $2x - y = 2$, find the value of $2x^2 + 5xy - 3y^2$.\n\nNow it's your turn to respond for the current step.\nYou should first conduct a reasoning process. This ... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"10"
]
}
} | {
"index": 74,
"source_id": "numinamath-277",
"split": "train"
} | Given $x + 3y = 5$ and $2x - y = 2$, find the value of $2x^2 + 5xy - 3y^2$. | - Recognize that the target expression depends on both variables, while the given constraints form a system of two linear equations; the immediate goal is to determine the unique ordered pair (x, y).
- Solve the linear system by eliminating one variable, then back-substitute to find the remaining variable's value.
- Ch... |
toolelite_math | [
{
"role": "user",
"content": "You are an expert agent tasked with answering the given question step-by-step.\nYour question: Given a positive integer \\( a \\), define a sequence of integers \\( x_1, x_2, x_3, \\ldots \\) such that \\( x_1 = a \\) and \\( x_{n+1} = 2x_n + 1 \\). Let \\( y_n = 2^{x_n} - 1 \\... | math-tool-reasoning | {
"style": "rule",
"ground_truth": {
"target": [
"2"
]
}
} | {
"index": 75,
"source_id": "numinamath-282",
"split": "train"
} | Given a positive integer \( a \), define a sequence of integers \( x_1, x_2, x_3, \ldots \) such that \( x_1 = a \) and \( x_{n+1} = 2x_n + 1 \). Let \( y_n = 2^{x_n} - 1 \). Determine the largest possible integer \( k \) such that for some positive integer \( a \), the numbers \( y_1, y_2, \ldots, y_k \) are all prime... | - Unroll the recurrence for x_n from the initial value a to obtain a closed-form expression for x_n in terms of n and a, so each exponent can be computed directly.
- Rewrite each y_n = 2^(x_n) - 1 as a Mersenne-type number with that exponent, allowing primality to be tested term by term.
- For a fixed candidate a, comp... |
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rl_trace_opsd
The versioned mathematical RL data used by the ToolElite TRACE-OPSD project.
The source contains 20,161 examples, with columns id, problem,
ground_truth, difficulty, hindsight, and solution. Difficulty labels are
mid (10,940 examples) and high (9,221 examples).
Two representations of the same examples
| Configuration | Split | Examples | Purpose |
|---|---|---|---|
source |
full |
20,161 | Full problem pool, including held-out validation IDs |
grpo |
train |
19,905 | Immutable training split in ToolElite/verl format |
grpo |
validation |
256 | Fixed training-monitoring validation split |
These are different representations of the same pool, not additional datasets.
Do not train on the source/full split without excluding the fixed validation
IDs. The validation split has 160 mid + 96 high examples; split seed is 42.
It is separate from final GSM8K/MATH500/AIME benchmark evaluation.
from datasets import load_dataset
training = load_dataset("BitZone910/rl_trace_opsd", "grpo", split="train")
validation = load_dataset("BitZone910/rl_trace_opsd", "grpo", split="validation")
source = load_dataset("BitZone910/rl_trace_opsd", "source", split="full")
The GRPO format includes the tool prompt, terminal answer reward, source IDs, and training-only privileged hindsight. At inference/evaluation, expose only the problem and tool prompt to the policy; do not leak answer or hindsight fields.
Source and attribution
Derived from NuminaMath-TIR, whose publisher declares Apache-2.0. ToolElite adds difficulty labels, solution hindsight, filtering, and tool-training formatting. Preserve upstream attribution and the included Apache license when redistributing.
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Models trained or fine-tuned on BitZone910/rl_trace_opsd